Diameter and Girth of Zero Divisor Graph of Multiplicative Lattices

Abstract

In this paper, we study the zero divisor graph ${\Gamma }^m(L)$ of a multiplicative lattice L. We prove under certain conditions that for a reduced multiplicative lattice L having more than two minimal prime elements, ${\Gamma }^m(L)$ contains a cycle and $gr({\Gamma }^m(L))=3$. This essentially proves that for a reduced ring R with more than two minimal primes, $gr(\mathbb{AG}(R)))=3$ which settles the conjecture of Behboodi and Rakeei [9]. Further, we have characterized the diameter of ${\Gamma }^m(L)$.